metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.12D10, (C2×C20)⋊20D4, C23⋊2(C4×D5), C10.94(C4×D4), (C2×Dic5)⋊16D4, C10.38C22≀C2, D10⋊3(C22⋊C4), C2.2(C20⋊2D4), C22.101(D4×D5), (C22×C4).31D10, C2.6(D10⋊D4), C2.3(C23⋊D10), C10.31(C4⋊D4), (C22×D5).125D4, C5⋊3(C23.23D4), C22.53(C4○D20), (C23×C10).39C22, C23.283(C22×D5), C10.10C42⋊39C2, C2.27(Dic5⋊4D4), C2.7(D10.12D4), C22.48(D4⋊2D5), (C22×C20).344C22, (C22×C10).330C23, (C23×D5).100C22, C10.32(C22.D4), (C22×Dic5).43C22, (C2×C5⋊D4)⋊9C4, C2.9(C4×C5⋊D4), (C2×C22⋊C4)⋊3D5, (D5×C22×C4)⋊13C2, (C2×C4)⋊12(C5⋊D4), (C2×Dic5)⋊7(C2×C4), (C2×C23.D5)⋊3C2, C2.29(D5×C22⋊C4), (C2×D10⋊C4)⋊4C2, (C10×C22⋊C4)⋊22C2, C22.127(C2×C4×D5), (C22×C10)⋊13(C2×C4), (C2×C10).322(C2×D4), C10.69(C2×C22⋊C4), (C22×C5⋊D4).2C2, C22.51(C2×C5⋊D4), (C22×D5).78(C2×C4), (C2×C10).145(C4○D4), (C2×C10).210(C22×C4), SmallGroup(320,583)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.12D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 1118 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23.23D4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C10.10C42, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, D5×C22×C4, C22×C5⋊D4, C24.12D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D5, C5⋊D4, C22×D5, C23.23D4, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×C22⋊C4, Dic5⋊4D4, D10.12D4, D10⋊D4, C4×C5⋊D4, C23⋊D10, C20⋊2D4, C24.12D10
►On 160 points
Generators in S160
(1 133)(2 59)(3 135)(4 41)(5 137)(6 43)(7 139)(8 45)(9 121)(10 47)(11 123)(12 49)(13 125)(14 51)(15 127)(16 53)(17 129)(18 55)(19 131)(20 57)(21 144)(22 89)(23 146)(24 91)(25 148)(26 93)(27 150)(28 95)(29 152)(30 97)(31 154)(32 99)(33 156)(34 81)(35 158)(36 83)(37 160)(38 85)(39 142)(40 87)(42 77)(44 79)(46 61)(48 63)(50 65)(52 67)(54 69)(56 71)(58 73)(60 75)(62 122)(64 124)(66 126)(68 128)(70 130)(72 132)(74 134)(76 136)(78 138)(80 140)(82 116)(84 118)(86 120)(88 102)(90 104)(92 106)(94 108)(96 110)(98 112)(100 114)(101 143)(103 145)(105 147)(107 149)(109 151)(111 153)(113 155)(115 157)(117 159)(119 141)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 81)(17 82)(18 83)(19 84)(20 85)(21 60)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 101)(41 136)(42 137)(43 138)(44 139)(45 140)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 45 31 55)(22 54 32 44)(23 43 33 53)(24 52 34 42)(25 41 35 51)(26 50 36 60)(27 59 37 49)(28 48 38 58)(29 57 39 47)(30 46 40 56)(61 74 71 64)(62 63 72 73)(65 70 75 80)(66 79 76 69)(67 68 77 78)(81 90 91 100)(82 99 92 89)(83 88 93 98)(84 97 94 87)(85 86 95 96)(101 131 111 121)(102 140 112 130)(103 129 113 139)(104 138 114 128)(105 127 115 137)(106 136 116 126)(107 125 117 135)(108 134 118 124)(109 123 119 133)(110 132 120 122)(141 142 151 152)(143 160 153 150)(144 149 154 159)(145 158 155 148)(146 147 156 157)
G:=sub<Sym(160)| (1,133)(2,59)(3,135)(4,41)(5,137)(6,43)(7,139)(8,45)(9,121)(10,47)(11,123)(12,49)(13,125)(14,51)(15,127)(16,53)(17,129)(18,55)(19,131)(20,57)(21,144)(22,89)(23,146)(24,91)(25,148)(26,93)(27,150)(28,95)(29,152)(30,97)(31,154)(32,99)(33,156)(34,81)(35,158)(36,83)(37,160)(38,85)(39,142)(40,87)(42,77)(44,79)(46,61)(48,63)(50,65)(52,67)(54,69)(56,71)(58,73)(60,75)(62,122)(64,124)(66,126)(68,128)(70,130)(72,132)(74,134)(76,136)(78,138)(80,140)(82,116)(84,118)(86,120)(88,102)(90,104)(92,106)(94,108)(96,110)(98,112)(100,114)(101,143)(103,145)(105,147)(107,149)(109,151)(111,153)(113,155)(115,157)(117,159)(119,141), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,81)(17,82)(18,83)(19,84)(20,85)(21,60)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,121)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,133), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,101)(41,136)(42,137)(43,138)(44,139)(45,140)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,45,31,55)(22,54,32,44)(23,43,33,53)(24,52,34,42)(25,41,35,51)(26,50,36,60)(27,59,37,49)(28,48,38,58)(29,57,39,47)(30,46,40,56)(61,74,71,64)(62,63,72,73)(65,70,75,80)(66,79,76,69)(67,68,77,78)(81,90,91,100)(82,99,92,89)(83,88,93,98)(84,97,94,87)(85,86,95,96)(101,131,111,121)(102,140,112,130)(103,129,113,139)(104,138,114,128)(105,127,115,137)(106,136,116,126)(107,125,117,135)(108,134,118,124)(109,123,119,133)(110,132,120,122)(141,142,151,152)(143,160,153,150)(144,149,154,159)(145,158,155,148)(146,147,156,157)>;
G:=Group( (1,133)(2,59)(3,135)(4,41)(5,137)(6,43)(7,139)(8,45)(9,121)(10,47)(11,123)(12,49)(13,125)(14,51)(15,127)(16,53)(17,129)(18,55)(19,131)(20,57)(21,144)(22,89)(23,146)(24,91)(25,148)(26,93)(27,150)(28,95)(29,152)(30,97)(31,154)(32,99)(33,156)(34,81)(35,158)(36,83)(37,160)(38,85)(39,142)(40,87)(42,77)(44,79)(46,61)(48,63)(50,65)(52,67)(54,69)(56,71)(58,73)(60,75)(62,122)(64,124)(66,126)(68,128)(70,130)(72,132)(74,134)(76,136)(78,138)(80,140)(82,116)(84,118)(86,120)(88,102)(90,104)(92,106)(94,108)(96,110)(98,112)(100,114)(101,143)(103,145)(105,147)(107,149)(109,151)(111,153)(113,155)(115,157)(117,159)(119,141), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,81)(17,82)(18,83)(19,84)(20,85)(21,60)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,121)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,133), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,101)(41,136)(42,137)(43,138)(44,139)(45,140)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,45,31,55)(22,54,32,44)(23,43,33,53)(24,52,34,42)(25,41,35,51)(26,50,36,60)(27,59,37,49)(28,48,38,58)(29,57,39,47)(30,46,40,56)(61,74,71,64)(62,63,72,73)(65,70,75,80)(66,79,76,69)(67,68,77,78)(81,90,91,100)(82,99,92,89)(83,88,93,98)(84,97,94,87)(85,86,95,96)(101,131,111,121)(102,140,112,130)(103,129,113,139)(104,138,114,128)(105,127,115,137)(106,136,116,126)(107,125,117,135)(108,134,118,124)(109,123,119,133)(110,132,120,122)(141,142,151,152)(143,160,153,150)(144,149,154,159)(145,158,155,148)(146,147,156,157) );
G=PermutationGroup([[(1,133),(2,59),(3,135),(4,41),(5,137),(6,43),(7,139),(8,45),(9,121),(10,47),(11,123),(12,49),(13,125),(14,51),(15,127),(16,53),(17,129),(18,55),(19,131),(20,57),(21,144),(22,89),(23,146),(24,91),(25,148),(26,93),(27,150),(28,95),(29,152),(30,97),(31,154),(32,99),(33,156),(34,81),(35,158),(36,83),(37,160),(38,85),(39,142),(40,87),(42,77),(44,79),(46,61),(48,63),(50,65),(52,67),(54,69),(56,71),(58,73),(60,75),(62,122),(64,124),(66,126),(68,128),(70,130),(72,132),(74,134),(76,136),(78,138),(80,140),(82,116),(84,118),(86,120),(88,102),(90,104),(92,106),(94,108),(96,110),(98,112),(100,114),(101,143),(103,145),(105,147),(107,149),(109,151),(111,153),(113,155),(115,157),(117,159),(119,141)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,81),(17,82),(18,83),(19,84),(20,85),(21,60),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,121),(109,122),(110,123),(111,124),(112,125),(113,126),(114,127),(115,128),(116,129),(117,130),(118,131),(119,132),(120,133)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,101),(41,136),(42,137),(43,138),(44,139),(45,140),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(81,157),(82,158),(83,159),(84,160),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,45,31,55),(22,54,32,44),(23,43,33,53),(24,52,34,42),(25,41,35,51),(26,50,36,60),(27,59,37,49),(28,48,38,58),(29,57,39,47),(30,46,40,56),(61,74,71,64),(62,63,72,73),(65,70,75,80),(66,79,76,69),(67,68,77,78),(81,90,91,100),(82,99,92,89),(83,88,93,98),(84,97,94,87),(85,86,95,96),(101,131,111,121),(102,140,112,130),(103,129,113,139),(104,138,114,128),(105,127,115,137),(106,136,116,126),(107,125,117,135),(108,134,118,124),(109,123,119,133),(110,132,120,122),(141,142,151,152),(143,160,153,150),(144,149,154,159),(145,158,155,148),(146,147,156,157)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 | D4×D5 | D4⋊2D5 |
| kernel | C24.12D10 | C10.10C42 | C2×D10⋊C4 | C2×C23.D5 | C10×C22⋊C4 | D5×C22×C4 | C22×C5⋊D4 | C2×C5⋊D4 | C2×Dic5 | C2×C20 | C22×D5 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 8 | 8 | 8 | 6 | 2 |
Matrix representation of C24.12D10 ►in GL6(𝔽41)
| 23 | 6 | 0 | 0 | 0 | 0 |
| 35 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 6 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 18 |
| 0 | 0 | 0 | 0 | 17 | 8 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 28 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 9 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 32 | 32 |
G:=sub<GL(6,GF(41))| [23,35,0,0,0,0,6,18,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,0,0,0,0,33,17,0,0,0,0,18,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[13,28,0,0,0,0,13,9,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[13,9,0,0,0,0,13,28,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;
C24.12D10 in GAP, Magma, Sage, TeX
C_2^4._{12}D_{10} % in TeX
G:=Group("C2^4.12D10"); // GroupNames label
G:=SmallGroup(320,583);
// by ID
G=gap.SmallGroup(320,583);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations